March  2019, 39(3): 1545-1558. doi: 10.3934/dcds.2019067

Liouville's theorem for a fractional elliptic system

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi 710129, China

* Corresponding author: Pengcheng Niu

Received  January 2018 Revised  May 2018 Published  December 2018

Fund Project: The authors are supported by the National Natural Science Foundation of China (No.11771354), China Postdoctoral Science Foundation (No.2017M613193)and Excellent Doctorate Cultivating Foundation of Northwestern Polytechnical University

In this paper, we investigate the following fractional elliptic system
$\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{\alpha /2}}u(x) = f(x){{v}^{q}}(x),&x\in {{R}^{n}}, \\ {{(-\Delta )}^{\beta /2}}v(x) = h(x){{u}^{p}}(x),&x\in {{R}^{n}}, \\\end{array} \right.$
where $1≤p, q < ∞$, $0 < α, β < 2$, $f(x)$ and $h(x)$ satisfy suitable conditions. Applying the method of moving planes, we prove monotonicity without any decay assumption at infinity. Furthermore, if $ α = β$, a Liouville theorem is established.
Citation: Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067
References:
[1]

W. AoJ. Wei and W. Yang, Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials, Discrete Contin. Dyn. Syst., 37 (2017), 5561-5601. doi: 10.3934/dcds.2017242. Google Scholar

[2]

F. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations, 70 (1987), 349-365. doi: 10.1016/0022-0396(87)90156-2. Google Scholar

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J. Bertoin, Lévy Processes, Cambridge Tracts in Mathmatics, 121 Cambridge University Press, Cambridge, 1996. Google Scholar

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C. BrandleE. ColoradoA. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. of Edinburgh, 143 (2013), 39-71. doi: 10.1017/S0308210511000175. Google Scholar

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X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar

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L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. Google Scholar

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W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure Appl. Anal., 12 (2011), 2497-2514. doi: 10.3934/cpaa.2013.12.2497. Google Scholar

[9]

W. Chen and C. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758. doi: 10.1016/j.aim.2018.07.016. Google Scholar

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W. ChenC. Li and Y. Li, A drirect method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038. Google Scholar

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W. Chen, Y. Li and P. Ma, The fractional Laplacian, in press.Google Scholar

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W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. Google Scholar

[13]

W. Chen, C. Li and J. Zhu, Fractional equations with indefinite nonlinearities, accepted by Discrete Contin. Dyn. Syst.Google Scholar

[14]

T. Cheng, Monotonicity and symmetry of solutions to frac- tional Laplacian equation, Discrete Contin. Dyn. Syst., 37 (2017), 3587-3599. doi: 10.3934/dcds.2017154. Google Scholar

[15]

C. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95. doi: 10.1007/BF00250684. Google Scholar

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P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical foundation of turbulent viscous flows, Lecture Notes in Math., 1871 (2004), 1-43. doi: 10.1007/11545989_1. Google Scholar

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Z. Dai, L. Cao and P. Wang, Liouville type theorems for the system of fractional nonlinear equations in $R^n_+$, J. Inequal. Appl., (2016), Paper No. 267, 17 pp. doi: 10.1186/s13660-016-1207-9. Google Scholar

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J. Dou and Y. Li, Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space, Discrete Contin. Dyn. Syst., 38 (2018), 3939-3953. doi: 10.3934/dcds.2018171. Google Scholar

[19]

D. FigueiredoP. Lions and R. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63. Google Scholar

[20]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196. Google Scholar

[21]

H. Kaper and M. Kwong, Uniqueness of non-negative solutions of a class of semilinear elliptic equations, Nonlinear Diffusion Equations and Their Equilibrium States Ⅱ, 13 (1988), 1-17. doi: 10.1007/978-1-4613-9608-6_1. Google Scholar

[22]

E. Leite ang M. Montenegro, On positive viscosity solutions of fractional Lane-Emden systems, 2015, arXiv:1509.01267.Google Scholar

[23]

Y. Li and P. Ma, Symmetry of solutions for a fractional system, Sci. China Math., 60 (2017), 1805-1824. doi: 10.1007/s11425-016-0231-x. Google Scholar

[24]

K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in $ {R}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145. doi: 10.1007/BF00275874. Google Scholar

[25]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2011), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar

[26]

P. NiuL. Wu and X. Ji, Positive solutions to nonlinear systems involving fully nonlinear fractional operators, Fract. Calc. Appl. Anal., 21 (2018), 552-574. doi: 10.1515/fca-2018-0030. Google Scholar

[27]

P. Pucci and V. Radulescu, The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., 3 (2010), 543-584. Google Scholar

[28]

A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659. doi: 10.1007/s00526-014-0727-8. Google Scholar

[29]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. Google Scholar

[30]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publicacions Matematiques, 58 (2014), 133-154. doi: 10.5565/PUBLMAT_58114_06. Google Scholar

[31]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445. Google Scholar

[32]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. Google Scholar

[33]

V. TarasovG. Zaslavsky and M. George, Fractional dynamics of systems with long-range interaction, Commun. Nonlinear Sci. Numer. Simul., 11 (2006), 885-898. doi: 10.1016/j.cnsns.2006.03.005. Google Scholar

[34]

P. Wang and P. Niu, A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pure Appl. Anal., 16 (2017), 1707-1718. doi: 10.3934/cpaa.2017082. Google Scholar

[35]

L. Wu and P. Niu, Symmetry and Nonexistence of Positive Solutions to Fractional p-Laplacian Equations, to appeared in Discrete Contin. Dyn. Syst., 2018.Google Scholar

[36]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125. Google Scholar

show all references

References:
[1]

W. AoJ. Wei and W. Yang, Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials, Discrete Contin. Dyn. Syst., 37 (2017), 5561-5601. doi: 10.3934/dcds.2017242. Google Scholar

[2]

F. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations, 70 (1987), 349-365. doi: 10.1016/0022-0396(87)90156-2. Google Scholar

[3]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathmatics, 121 Cambridge University Press, Cambridge, 1996. Google Scholar

[4]

C. BrandleE. ColoradoA. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. of Edinburgh, 143 (2013), 39-71. doi: 10.1017/S0308210511000175. Google Scholar

[5]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[6]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar

[7]

L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. Google Scholar

[8]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure Appl. Anal., 12 (2011), 2497-2514. doi: 10.3934/cpaa.2013.12.2497. Google Scholar

[9]

W. Chen and C. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758. doi: 10.1016/j.aim.2018.07.016. Google Scholar

[10]

W. ChenC. Li and Y. Li, A drirect method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038. Google Scholar

[11]

W. Chen, Y. Li and P. Ma, The fractional Laplacian, in press.Google Scholar

[12]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. Google Scholar

[13]

W. Chen, C. Li and J. Zhu, Fractional equations with indefinite nonlinearities, accepted by Discrete Contin. Dyn. Syst.Google Scholar

[14]

T. Cheng, Monotonicity and symmetry of solutions to frac- tional Laplacian equation, Discrete Contin. Dyn. Syst., 37 (2017), 3587-3599. doi: 10.3934/dcds.2017154. Google Scholar

[15]

C. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95. doi: 10.1007/BF00250684. Google Scholar

[16]

P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical foundation of turbulent viscous flows, Lecture Notes in Math., 1871 (2004), 1-43. doi: 10.1007/11545989_1. Google Scholar

[17]

Z. Dai, L. Cao and P. Wang, Liouville type theorems for the system of fractional nonlinear equations in $R^n_+$, J. Inequal. Appl., (2016), Paper No. 267, 17 pp. doi: 10.1186/s13660-016-1207-9. Google Scholar

[18]

J. Dou and Y. Li, Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space, Discrete Contin. Dyn. Syst., 38 (2018), 3939-3953. doi: 10.3934/dcds.2018171. Google Scholar

[19]

D. FigueiredoP. Lions and R. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63. Google Scholar

[20]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196. Google Scholar

[21]

H. Kaper and M. Kwong, Uniqueness of non-negative solutions of a class of semilinear elliptic equations, Nonlinear Diffusion Equations and Their Equilibrium States Ⅱ, 13 (1988), 1-17. doi: 10.1007/978-1-4613-9608-6_1. Google Scholar

[22]

E. Leite ang M. Montenegro, On positive viscosity solutions of fractional Lane-Emden systems, 2015, arXiv:1509.01267.Google Scholar

[23]

Y. Li and P. Ma, Symmetry of solutions for a fractional system, Sci. China Math., 60 (2017), 1805-1824. doi: 10.1007/s11425-016-0231-x. Google Scholar

[24]

K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in $ {R}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145. doi: 10.1007/BF00275874. Google Scholar

[25]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2011), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar

[26]

P. NiuL. Wu and X. Ji, Positive solutions to nonlinear systems involving fully nonlinear fractional operators, Fract. Calc. Appl. Anal., 21 (2018), 552-574. doi: 10.1515/fca-2018-0030. Google Scholar

[27]

P. Pucci and V. Radulescu, The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., 3 (2010), 543-584. Google Scholar

[28]

A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659. doi: 10.1007/s00526-014-0727-8. Google Scholar

[29]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. Google Scholar

[30]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publicacions Matematiques, 58 (2014), 133-154. doi: 10.5565/PUBLMAT_58114_06. Google Scholar

[31]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445. Google Scholar

[32]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. Google Scholar

[33]

V. TarasovG. Zaslavsky and M. George, Fractional dynamics of systems with long-range interaction, Commun. Nonlinear Sci. Numer. Simul., 11 (2006), 885-898. doi: 10.1016/j.cnsns.2006.03.005. Google Scholar

[34]

P. Wang and P. Niu, A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pure Appl. Anal., 16 (2017), 1707-1718. doi: 10.3934/cpaa.2017082. Google Scholar

[35]

L. Wu and P. Niu, Symmetry and Nonexistence of Positive Solutions to Fractional p-Laplacian Equations, to appeared in Discrete Contin. Dyn. Syst., 2018.Google Scholar

[36]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125. Google Scholar

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